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Statistics > Methodology

Abstract: Given a causal graph, the do-calculus can express treatment effects as
functionals of the observational joint distribution that can be estimated
empirically. Sometimes the do-calculus identifies multiple valid formulae,
prompting us to compare the statistical properties of the corresponding
estimators. For example, the backdoor formula applies when all confounders are
observed and the frontdoor formula applies when an observed mediator transmits
the causal effect. In this paper, we investigate the over-identified scenario
where both confounders and mediators are observed, rendering both estimators
valid. Addressing the linear Gaussian causal model, we derive the finite-sample
variance for both estimators and demonstrate that either estimator can dominate
the other by an unbounded constant factor depending on the model parameters.
Next, we derive an optimal estimator, which leverages all observed variables to
strictly outperform the backdoor and frontdoor estimators. We also present a
procedure for combining two datasets, with confounders observed in one and
mediators in the other. Finally, we evaluate our methods on both simulated data
and the IHDP and JTPA datasets.